Talk:Cascading-E notation
If BEAF is in Functions category, should this and Hyper-E notations be there too? Seems like a good idea. Deedlit11 (talk) 10:01, April 6, 2013 (UTC) I think the limit growth rate of Cascading-E notation can be far beyond ε0.At the end of this page , Saibian came up with something like E100#^^##100, E100#^^^#100, E100#^^^^#100, E100#^^^^^#100, E100{#,#,#}100 (using BEAF), E100{#,#(1)2}100, E100{#,#(2)2}100, E100{#,#((((0,1)1)1)1)2}100, E100{10^^^100&#}100, E100{10^^^^100&#}100, E100{#&#&#&#&#&#&#&#}100, etc. That makes it as powerful as BEAF. However, that isn't the actual growth rate of Cascading-E notation because we use BEAF in it. How about the growth rate without using BEAF? Here is some of my ideas. Ea#^^#b can be written as Ea f(#) b, where f(x)=x^^x=E(x)1#1#2. Further, f(#) can be any recusive function whose "base number" is #, such as f(x)=E(x)1#1#1#1#1#1#2=x^^^^^^x, f(x)=E(x)x##x, f(x)=E(x)x##x#1#2, f(x)=E(x)x##x##x, f(x)=E(x)x###x, f(x)=E(x)x#######x, f(x)=E(x)x#^#x, f(x)=E(x)x#^##x, f(x)=E(x)x#^#^#x, f(x)=E(x)x#^#^#^#^#^#^#^#x, etc. Then f(x)=E(x)x g(#) x, where g(#) can be any recusive function whose "base number" is #, such as g(x)=E(x)x#x#x#x, g(x)=E(x)x##x, g(x)=E(x)x###x#x, g(x)=E(x)x#^#x, g(x)=E(x)x#^#^#^#^#^#x, and g(x)=E(x)x h(#) x, where h(#) can be any recusive function whose "base number" is #. I think, Ea f(#) b has comparable growth rate to f(n)&n, or g(n)&n&n, or h(n)&n&n&n. Just imagine this function: En f1(#) n, where f1(x)=E(x)x f2(#) x, where f2(x)=E(x)x f3(#) x, ... where fm(x)=x^x. The m here is Cascading-E notation nests. It has comparable growth rate to n^n&n&n&n...&n (with m &'s). If m=n, we'll get the actual limit growth rate of Cascading-E notation --- the LVO. {hypcos} (talk) 04:05, July 13, 2013 (UTC) :I believe it's good extension, but we need more formalization on it. Also, it's highly dependent on what you defined increment to f(n). If Ea f(n+1) b = Ea f(n) a f(n+1) b-1, then your idea will be about notation up to \(\Gamma_0\) range (and I already wrote about such extension it in my blog post). If, however, Ea f(n)+1 b = Ea f(n) a f(n)+1 b, then it really can reach LVO. The difference is big, as in the first case we increment the argument of function in order to get "new type" of hyperion-array, and in another way, we increment the result of function. Ikosarakt1 (talk ^ ) 10:16, July 13, 2013 (UTC) :: The rule is that @a f(#)*# b=@a f(#) a f(#)*# (b-1), where @ represent any sequence in Cascading-E notation. {hypcos} (talk) 15:31, July 13, 2013 (UTC) Extended Cascading-E Notation Ea#^^##b = Ea(#^^#)^(#^^#)...(#^^#)^(#^^#)a (b-1 ^'s) Ea#^^#^#b = Ea#^^##...##a (b #'s after ^^'s) Ea#^^#^^#b = Ea#^^#^#...#^# (b-1 ^'s after ^^'s) Ea#^^^#b = Ea#^^#...#^^#a (b-1 ^^'s) Ea#^^^##b = Ea(#^^^#)^^^#b Tethrathoth = E100#^^#100 Tethrathothtethrathothex = E100#^^#100#2 Tetrithrathoth = E100#^^#*#3 Teterthrathoth = E100#^^#*#4 Tetppegathrathoth = E100#^^#*#5 Tetgoldthrathoth = E100#^^#*#100 Tethroothrathoth = E100#^^#*##100 Deutero-tethrathoth = E100#^^#*#^^#100 Hecato-tethrathoth = E100(#^^#)^#100 Todthrathoth = E100(#^^#)^(#^^#)100 Todthrathor = E100(#^^#)^(#^^#)^(#^^#)100 Territhrathoth = E100#^^##100 Treathrathoth = E100#^^###100 Tiganthrathoth = E100#^^#^#4 Torgethrathoth = E100#^^#^#5 Tugolthrathoth = E100#^^#^#100 Tethrathor = E100#^^#^^#100 Penthrathoth = E100#^^^#100 Hexthrathoth = E100#^^^^#100 AarexTiao 01:44, January 6, 2014 (UTC) LECEN Sbiis Saibian changed the definiton of tethrathoth-ex-terrible tethrathoth to E100#^^#>#(1+E100#^^#>100). So this might be the beginning of LECEN. Wythagoras (talk) 17:32, January 16, 2014 (UTC) :Actually, LECEN (Limited Extension Cascading-E Notation) is the notation that goes up to the Great and Terrible Tethrathoth, what Saibian already revealed. The new notation you probably meant is "Extended Cascading-E Notation" (xE^), which will go even farther than that. So hype! -- ☁ I want more ⛅ 02:22, January 17, 2014 (UTC) ::[https://sites.google.com/site/largenumbers/home/4-3/extended_cascading-e The next next level has been unlocked.] -- ☁ I want more ⛅ 05:58, January 31, 2014 (UTC) :::I haven't expected this... Ikosarakt1 (talk ^ ) 21:34, January 31, 2014 (UTC) @Littlepeng9 Yup. E#, xE#, E^, xE^, and ExE all should be "Functions" along with being "Notations". UAN under construction 08:28, February 16, 2015 (UTC)